Solutions in

Institut Tiir PhysikaIische und Theoretische Chemie der Universitat Regensburg, F R G

in Chemistry and Modern Technology

Josef Barthel, Heiner-J. Gores, Georg Schmeer, and Rudolf Wachter

Table of Contents

Part A

Fundamentals of Chemistry and Physical Chemistry of Non-Aqueous Electrolyte Solutions

I Introduction 37 II Classification of Solvents and Electrolytes 38

III Thermodynamics of Electrolyte Solutions 40 IV Short and Long-Range Forces in Dilute Electrolyte Solutions 43

4.1 DistributionFunctionsandMean-ForcePotentials 43 4.2 The Basic Chemical Model of Electrolyte Solutions 44

4.3 The Ion-Pair Concept 46 V Thermodynamic Properties of Electrolyte Solutions 48

5.1 Generalities 48 5.2 Solution and Dilution Experiments 49

5.3 EMF-Measurements 52 5.4 Some Remarks on Thermodynamic Investigations 53

5.5 Ion Solvation 54 5.6 Concentrated Solutions 56

5.7 Water at Extreme External Conditions 57

VI TransportProperties 57 6.1 Dilute Solutions 57 6.2 Concentrated Solutions 61 6.3 Ion Aggregates and their Role in Conductance 62

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VII Spectroscopic and Relaxation Methods 64

7.1 Introduction 64 7.2 Diffraction Methods 65

7.3 Absorption Spectroscopy 67 7.4 N M R and Related Methods 70

7.5 ESR-Spectroscopy 71 7.6 Relaxation Methods 71 VIII Chemical Reactions in Organic Solvents 74

8.1 Ions and Ion Pairs in Chemical Reactions , 74

8.2 Kinetic Solvent Effects 75 8.3 The Use of Correlation Functions 80

8.4 Phase-Transfer Reactions 82

Part B

Technical Applications and Applied Research

Univ.-Biblioihek,

Rcpensktrg j IX Introduction 83 X High-Energy Batteries 85

10.1 Background 85 10.2 Non-Aqueous Electrolyte Solutions in Lithium Batteries 87

10.3 Stability of Electrolyte Solutions with Lithium 90

10.4 Non-Aqueous Primary Cells 91 10.4.1 Commercial Cells with Solid Cathodes and Organic Solvents 91

10.4.2 Commercial Cells with Liquid Cathodic Materials . . . . 91

10.4.3 RecentDevelopments 92

10.5 Secondary Batteries 93 10.5.1 Improvements of the Cycling Efficiency of the Anode . . . 93

10.5.2 Cathode Materials for Secondary Lithium Batteries. . . . 95

XI Non-Emissive Electro-Optic Displays 96 11.1 Comparison of Methods 96 11.2 DisplayswithNon-AqueousElectrolyteSolutions 97

XII Photo-Electrochemical Cells 98

12.1 Introduction 98 12.2 Liquid-Junction Cells 99

12.3 Non-Aqueous Solutions in Liquid-Junction Cells 100 XIII Electrodeposition W 2

13.1 Introduction

13.2 Selected Examples 103 XIV Wet Electrolytic Capacitors 104

X V Klectro-Organic Synthesis 105 15.1 DisappointmentsandAdvantages 105

15.2 The Influence of Electrolyte Solutions on Reactions 107 15.3 Selected Examples from Actual Investigations 108

XVI Further Promising Fields of Application 110 16.1 Processes Based on Solvating Properties 110

16.2 Electropolishing I l l 16.3 Anodic Oxidation of Semiconductors I l l

XVII Acknowledgements I l l XVIII Appendices I l l

A Solvent and Electrolyte Data I l l A . 1 Properties of Organic Solvents I l l

A . 2 IonDistanceParameters 116 B Electrostatic Potential around a Particle with

an Arbitrary Charge Distribution 116 C Electrostatic Potential around a PoIarisable ApoIar Particle . . . 119

D SymbolsandAbbreviations 121 D. 1 Fundamental Constants 121 D.2 Generally Used Superscripts and Subscripts 121

D.2.1 Superscripts 121 D.2.2 Subscripts 121

D.3 Symbols 122 XIX References(PartAandB) 123 XX References (Appendices A, B , C and D) 141

In this paper a brief survey is given of the properties of non-aqueous electrolyte solutions and their applications in chemistry and technology without going into the details of theory. Specific solvent-solute interactions and the role of the solvent beyond its function as a homogenous isotropic medium are stressed. Taking into account Parker's statement n "Scientists nowadays arc under increasing pressure to consider the relevance of their research, and rightly so" we have included examples showing the increasing industrial interest in non-aqueous electrolyte solutions.

The concepts and results are arranged in two parts. Part A concerns the fundamentals of thermodynamics, transport processes, spectroscopy and chemical kinetics of non-aqueous solutions and some applications in these fields. Part B describes their use in various technologies such as high-energy batteries, non-emissive electro-optic displays, photoelectrochemical cells, clcctrodeposi- tion. electrolytic capacitors, electro-organic synthesis, metallurgic processes and others.

Four Appendices are added. Appendix A gives a survey on the most important non-aqueous solvents, their physical properties and correlation parameters, and the commonly used abbreviations.

Appendices U and C show the mathematical background of the general chemical model. Ihe symbols and abbreviations of the text are listed and explained in Appendix D.

PartA

Fundamentals of Chemistry and Physical Chemistry of Non-Aqueous Electrolyte Solutions

I Introduction

The systematic investigation of non-aqueous solutions is guided by the progress of our knowledge on solute-solute and solute-solvent interactions. By combination with chemical models of the solution, valuable results can be obtained which assist the understanding of the properties of these solutions.

For Dilute Electrolyte Solutions consistent and reliable equations are based on the modern conception of electrochemistry which takes into account both long and short-range forces between the solute and solvent particles in the framework of the McMillan-Mayer-Friedman approaches to theory ^{2 )}. Solution chemists usually think of short-range interactions in terms of ion-pair formation. A chemical model of electrolyte solutions as developed and used in our laboratory ^{3 , 4 )} is the basis of the fundamentals given in part A of this survey. It allows the use of the classical asso- ciation concept initially introduced by Bjerrum ^{5 )} after some refinements concerning the spatial extension and structure of ion pairs and the mean-force potentials.

Classical thermodynamics and transport phenomena are unable to distinguish between ion pairs and undissociated electrolyte molecules, both proving their presence in the solutions as neutral particles in equilibrium with the Tree' ions.

However, in favourable cases ion pairs may be detected separately from undissociated electrolyte molecules by spectroscopic methods. The ions in an ion pair retain their individual ionic characters and are linked only by Coulombic and short-range forces, including H-bonding.

Modern developments of the statistical-mechanical theory of solutions provide valuable results, but no satisfactory answer can yet be found to fundamental questions such as the effect of ions on the permittivity of the solvent or on the structures in solution^{2 )}. Computer simulations may be helpful in understanding how some fundamental properties of the solutions are derived from fundamental laws. However, the actual limitation to a set of a few hundred particles in a box of about 20 A of length, a time scale of the order of picoseconds, and pair potentials based on classical mechanics usually prevent the deduction of useful relationships for the properties of real electrolyte solutions.

The treatment of Electrolyte Solutions from Moderate to Saturated Concentrations either rationalizes the effects from ion-ion and ion-solvent interactions in terms

of the parameters characterizing the behaviour of electrolytes in dilute solutions or uses correlation methods based on empirical interaction scales ^b ~^{2 n}. Examples of both procedures will be given.

II Classification of Solvents and Electrolytes

Any attempt to set up a complete theory which takes into account all types of interaction energy between the ions and molecules of an electrolyte solution with the aim of determining the properties of the solution by means of statistical thermo- dynamic methods must be unsuccessful as a result of insuperable mathematical difficulties. The approaches to the problem which have been used were outlined in Section 1.

Two aspects determine the role of the solvent: its bulk properties and its electron donor or acceptor abilities. The Debye-Huckel theory ^{2 2 )} which is valid at infinitely low concentrations, recognizes solvents only by their bulk properties, i.e. relative permittivity 8, viscosity r|, and density Q. However, the Debye-Huckel range of validity is often experimentally unattainable (Ref. ^{4 )}, cf. also Figs. 4 and 6). The importance of bulk properties decreases with increasing electrolyte concentration.

Donor and acceptor properties are the main factors which govern processes on the molecular scale, i.e. solvation and association. Theoretical and semipheno- menological approaches use molecular properties, dipole and quadrupole moments, polarizability etc., or mean-force potentials for taking these effects into conside- ration. Applied solution chemistry takes account of them with the help of the em- pirical scales previously mentioned ⁶ ~^{2 1 )}.

Various attempts have been made to classify solvents, e.g. according to bulk and molecular properties ^{2 0 )}, empirical solvent parameter s c a l e s⁶"^{2 1 1}, hydrogen-bonding ability 2 3 2 4 ), and miscibility ^{2 5 )}. In table I solvents are divided into classes on the basis of their acid-base properties ^{2 6}"^{2 9 )} which can be used as a general chemical measure of their ability to interact with other species. Detailed information on these and other solvents, their symbols, fusion and boiling points (9^{| ;} and O^u), bulk properties (£,r|, Q), and currently-used correlation parameters D N (donor number), R^rV a l u e , and A N (acceptor number) is given in Appendix A - I .

Table I Classification of organic solvents (for detailed information see: Appendix A-I)

Solvent class

1. amphiprotic hydroxylic 2. amphiprotic protogenic 3. prolophilic H-bond donor 4. aprotic protophilic

5. aprotic protophobic

(i. low permittivity electron donor

7. inert

Examples

methanol (MeOH); ethanol (EtOH); 1-propanol (PrOH); diethylene glycol (DEG); glycerol;

acetic acid

formamide (FA); N-methylformamide ( N M F ) : diaminoethane;

dimethylformamide ( D M F ) ; l-methyl-2-pyroli- done (NMP); hexamethylphosphoric triamide (HMPT); dimethylsulfoxidc (DMSO) acetonitrile (AN); sulfolane (TMS); propylene carbonate (PC); y-butyrolactone (y-Bl); acetic anhydride

diethyl ether (DEE); lctrahydrofuranc (TlIF);

diglyme(DG); 1,2-dimelhoxyethanc ( D M E ) ; 1,4- dioxane

dichloromethanc; IclrucliIornclIiylcnc, ben/enc, cyi Iohexanc

The symbols quoted there, e.g. M e O H , T H F , D M E etc., are used in this text.

No classification is universally applicable. Overlapping of the solvent classes is inevitable and some specific solute-solvent interactions evade classification. Specific interactions, however, are often sought i n connexion with technological problems and have led to a search for appropriate solvent mixtures which are gaining impor- tance in many fields of applied research. In spite of all its limitations, the classifi- cation of solvents is useful for rationalizing the choice of appropriate solvents and solvent mixtures for particular investigations.

Electrolytes can be classified in two categories, ionophores ^{3 0 )} (true electrolytes^{3 1 }}) and ionogenes ^{3 0 )} (potential electrolytes ^{3 1 ]}). Ionophores are substances which, in the pure state, already exist as ionic crystals, e.g. alkali metal halides. Ionogenes, such as carboxylic acids, form ions only by chemical reactions with solvent molecules.

Amphiprotic solvents themselves behave as ionogenes in producing their lyonium ions (anions) and lyate ions (cations) by autoprotolysis reactions. Ionophores are initially completely dissociated in solution and their ions are solvated. However, almost all solvents allow ion-association to ion pairs and higher ion aggregates, both with and without inclusion of solvent molecules, to occur. When electrically neutral these species cannot transport current. The following examples are given for illustration:

(a) ionophores:

NaCl(cryst) —> N a⁺ + C l " ^ T N a⁺C T ]⁰ LiBF⁴(Cryst) — - — — - L i⁺ + B F⁴; [ L i⁺B F⁴: ]⁰

•+ J 1 (limelhoxyethane **- L

and further asspciation to [Li ⁺ ( B F⁴^ )²] ' etc.

(b) ionogenes:

H C l O⁴ + C H³C O O H £ ± [ CH³C O O H⁺C I O⁴]⁰ ^ ± C H³C O O H i +

Kc + C l O⁴

C H³O H + C H³O H «=± C H³O H⁺ + C H³O " .

These association and dissociation reactions do not usually proceed to completion.

Both processes are described by the thermodynamic equilibrium constants K ^A (association constant) or K ^d (dissociation constant). The dissolution o f perchloric acid in glacial acetic a c i d^{3 2}"^{3 4}* shows the typical ionisation equilibrium (equili- brium constant, K¹) preceding the dissociation process in the case o f ionogenes.

The overall constant K is given by

K , = K , K , ) - . (1)

1 \ K¹

For strong acids and bases, where K¹ 'y I, equation (I) reduces to K ~ K^{l r} whereas Inr weak ones, where K¹ <| I, K K¹K^u. The sequence of basicities can change with changing solvent; the K^l-Values are more significant for the discussion of ionogenes than are the K-values. Ionisation constants, however, will not be discussed in this article. It is sufficient to note that the dissolution of acids or bases

in amphiprotic solvents is generally followed by a protolysis reaction (partial neu- tralisation) as a consequence of the following equilibria:

where S H represents the amphiprotic solvent, A H the acid and B the base. When these equilibria are shifted markedly towards ionisation, a levelling effect^{3 5 )} occurs which almost completely replaces the acids or bases with the lyate or lyonium ions of the solvent, irrespective of the initial strong acid or base. A typical example of this is the behaviour of the mineral acids in water.

The series of simultaneous equilibria including K , , K ^d (or K^A) and the auto- protolysis constant K ^S limits the quantitative discussion of electrolyte solutions to simple cases. However, the appropriate choice of such cases will give valuable insight into the properties of electrolyte solutions, especially those of ionophores where the ionisation step need not be considered.

Finally, a classification of the individual ions is only possible in a rough and incomplete way. Monoatomic cations can be arranged according to the number of electrons and are referred to as "dⁿ-cations" (e.g. alkali metal cations then arc d°- cations). Cations such as [ R⁴N ]⁺, [R⁴P] ⁺ , or [ R³S ]⁺ have their charges shielded by alkyl or aryl groups. They are almost non-polarizable and are referred to as inert cations. Cations of type [ R⁴_ⁿHⁿN ]⁺ or [ ( R O H )⁴N ]⁺ are protic cations, capable of forming H—bonds with anions or solvent molecules. The distinction between

"hard" and "soft*' cations has no significant relevance here, but for anions it is useful. A detailed discussion of the properties of electrolyte solutions as a function of ion classes and solvents is given in Ref. ^{3 6 )}.

It should be mentioned that experimental investigations, especially in dilute non-aqueous solutions, require highly purified solvents and solutes. Impurities can change the properties of the solution drastically. A water content of 20 ppm is equivalent to the total amount of solute in a IO"'³ molar solution. For checking the purity of solvents U V cut-off, conductivity, chromatography as well as thermal and electrochemical methods are recommended ^{3 7}~^{4 1}>. The control of the purity of electrolytes is more difficult; for details see Refs. ^{3 7}~^{4 2}>.

IH Thermodynamics of Electrolyte Solutions

The thermodynamic properties of an electrolyte solution can be derived from the chemical potentials U^j of its components which are given by the relationships^{4 3}'.

A H + S H *± A " + S H ⁺

B + S H *± B H⁺ + S" (2b)

(2 a)

cf. also

K ( P . T ) -- u*(p, T) + R T In x^sf^s; (3a)

u^s*(p, T) = Iim u^s(p, T); (3b, c)

for the solvent S, and

u.(p, T) = u?<^c>(p, T) + Vⁱ R T In c<±y»; i = 1, 2,... (4a)

u r^{( c )}( p , T) = Iim [u,.(p, T) - Vⁱ R T In c!?]; Iim y«> = 1 (4b, c)

for the electrolyte components Yⁱ = (X*¹)^ ( X^{z 2 2}) ^ (Xj³)^v^ ... which dissociate into their ions X^{2 a}"¹ according to the scheme Yⁱ -» £ VJⁿX^{2 n 1}".

m

In these equations, x^s and c^ are the mole fraction of the solvent, S, and the mean concentrations [mol/dm³ of solution] of the electrolytes, Yⁱ; f^s and y ^ are the appropriate activity coefficients

C ^ = F I C^{y v i}; y⁽i^, = l l y r^{/ V i}; ^ = K - ( 4 d , e , D m m rn

Eqs. (4a) are replaced by an equivalent set of equations

H.(p, T) = u f^{( m )} 4- V¹ R T In ; i = 1, 2,... (5) if mean molalities, m ^ [mol/kg of solvent] are chosen as the concentration scale

with as the appropriate activity coefficients. Eqs. (4b-f) then must be appro- priately changed. Conversion formulae for the reference chemical potentials, con- centration scales and activity coefficients are given in Refs. ³^^{3 6}'^{4 3}-^{4 4}) .

The relationships 3 and 4, or 3 and 5, respectively, form a complete set of equations on which a general discussion of the thermodynamic properties of the electrolyte solution can be based.

As only some fundamental ideas can be illustrated in this article, the following discussion is restricted to solutions of a solvent S and a symmetrical electrolyte Y = C^{z +} A^z~ yielding z⁺-valent cations C^{z +} and z_-valent anions A²" . The chemical potential of the solute is then given by the relationships

u^Y(p, T) = u?^{( c )}(p, T) + 2 R T In c^{± y ±} (6a)

c^± = [ c⁺c _ r² = c^Y = c ; y^± = [y ⁺ y j^{1 / 2} (6b, c) or by a similar set of equations derived from eq. (5).

Eq. (6 a) is valid for both completely and partially dissociated (or associated) electrolytes if the activity coefficient is written as follows

y± = ^ay'± (7)

with a as the degree of dissociation and y'^± as the activity coefficient of the dissociated part (free ions) of the electrolyte component of the solution ⁴^^{3 6}-^{4 4}* . For completely dissociated electrolytes a is equal to 1. The ion pairs of ionophoric electrolytes and the molecules of ionogenic electrolytes are in equilibrium with the free ions. The

equilibrium condition yields the equilibrium constant as an association ( K^a) or dissociation ( K^d) constant

K^A = K D¹ = —72 (8)

a cy±

The solution process of n^Y mol of electrolyte Y in n^s mol of solvent S is accompanied by a change in Gibbs energy

^^{5 0 1}G = n^s(M⁵ - H⁸*) + n^Y( ^^Y - n*). (9)

Analogous equations are obtained for every other extensive thermodynamic property Z , i.e.

A^{s o l}Z = n^s(Z^s - Z*) + n^Y( Z^Y - Z*) (10)

where Zⁱ and Zf are the partial molar quantities of the solvent or the electrolyte in the solution (Zⁱ) and in the pure phase (Z¹*). Another useful formulation O f A^{s o l}Z is based on the definition of apparent molar quantities O^z of the solute

Z = n,Z* + n^Y<D^z (11)

yielding the relationship

A- ^ ? = A^{s o l}Z - + (d)^z - « ? ) ; A^{s o l}Z - = Z * - Z * (12a, b)

QUANTITY ZY OPERATIONS QUANTITY Z Ions Cz*and Az-

in the vacuum

A 7so Iv*- Y ⁰⁰ A s o / Z "

A f^tZy

EtectrolyteY=Cz^Az

(pure phase) + Solvent S

Solution at AL

concentration c^Y

T T

+ Solvent S Solution at

concentration c'Y<cY

Aso/ Z

Ad, / Z

. 1

+ (ooJSolvenJ^ S *-(oo) Solvent S'

r AtrZ?

^Infinitely dilute solutior in the solvent St-Cy^O

Infinitely dilute solution

A

in the solvent S-Cy-^O

Fig. I. Operations on electrolyte solutions at constant pressure and temperature and their appropriate translation into thermodynamic quantities:

AZ = Zf i n -— Zi n i t

!at: lattice; solv: solvation; sol: solution; dil: dilution; tr; transfer from solvent S' to solvent S.

Z = V (volume); S (entropy); C^p (heat capacity); H (enthalpy): G (Gibbs energy)

if the thermodynamic relation ¢)¾ = Zy is taken into account. The molar quantity

A^{S O 1}Z^y corresponds to the transfer of 1 mol electrolyte compound from its pure

state to infinite dilution in the solvent S.

Figure 1 summarizes the operations and notions used in solution chemistry.

LV Short and Long-Range Forces in Dilute Electrolyte Solutions 4.1 Distribution Functions and Mean-Force Potentials

The statistical theory of electrolyte solutions is built up around the distribution func- tions of the ions, cf. ^{3 c}^^{4 4}-^{4 9}) . Electrolytes in solution give ions of the types

X^{1 Z i}, X²'², ... in the analytical concentrations N¹, N², ... ions/cm³. The distribution

of the ions in the solution depends on the forces acting between all the particles, ions and solvent molecules. External forces are also involved in the description of transport properties.

The analytical concentrations, N . , are one-particle molecular distribution func- tions and do not provide any information on particle interactions. Two-particle molecular distribution functions

indicate the probability of finding two ions, Xⁱ and X^j, simultaneously at points P¹(I^{t 1}) and P²(f²) in the solution, regardless of the position of the remaining ions and regardless of the velocities of all the particles, see Fig. 2. The pair-correlation

^ f¹, f²) = NiNjftjft, f²) = N j N^{l g j l}( r², r,) = fj,(f², r,) (13)

0 R

Fig. 2. The chemical model of electrolyte solutions. 0 : obser- ver, i: ion X . ; j : ion X^j in an arbitrary position, f^{2 1}, with re- gard to the ion X^j; special posi- tions (contact, separation by one or two orientated solvent molecules) are sketched with broken lines, r, a, R: distance parameters; W^{i j}: mean-force potentials; V^{i j} and v.,: relative velocities of ions X^j and Xⁱ

functions, g^fp f²), are related to the mean-force potentials W^{j j}( F ^ f²) between the ions Xⁱ and X^j by the relationship

W^{i j}C f p f²) = ~ k T In S^j( F p f²) . (14)

In addition they can be used to determine the local concentrations N^{i j} of ions X^j at a distance f^{2 1} (f^{2 1} = V² — T¹) from an ion X¹ situated at V¹

N ^ f p f ^ ^ N ^ f p f , , ) . (15) The relative velocities (see Fig. 2) of ions Xⁱ and X^j, v\.(F^p f^{2 1}) and v\.(f², f^{1 2}) ,

and the two-particle molecular distribution functions are linked by Onsager's continuity equation ^{4 5 )}:

where the differential operators are applied with regard to the coordinates in f, and f². A general treatment yielding E q . (16) starts from Liouville's theorem and uses the B B G K Y hierarchy of equations 4 8 > 5 2 ).

For basic information on electrolyte theory see Refs. 2 - 4 , 2 2 , 3 0 , 3 6 . 4 4 - 7 0 ^

4.2 The Basic Chemical Model of Electrolyte Solutions

In order to obtain a framework which allows the development of appropriate equations for the properties of solutions at low electrolyte concentrations, a model of the ions and their surroundings must be used which takes into account both short and long-range forces. For this purpose the space around an ion is subdivided into three regions (see Fig. 2).

i) r ^ a, a being the minimum distance of two oppositely-charged ions which is assumed to be the sum of effective cation and anion radii, a = a⁺ + a_.

ii) a ^ r ^ R, within which a paired state of oppositely-charged ions, the so-called ion pair, suppresses long-range interactions with other ions in the solution. In dilute solutions the occupation of the region a g r ^ R by ions of the same sign or by more than two ions can usually be neglected.

iii) r ^ R, the region of long-range ion-ion Coulombic interactions.

Table II shows the mean-force potentials for a dilute solution of the electrolyte compound Y = C^{2 +}A^z" .

The model is a McMillan-Mayer (MM)-Ievel Hamiltonian model. Friedman ^{2 )} characterizes models of this type as follows: ^{t i}W i t h MM-models it is interesting to see whether •one can get a model that economically and elegantly agrees with all of the relevant experimental data for a given system; success would mean that we can understand all of the observations in terms of solvent-averaged forces between the ions. However, it must be noted that there is no reason to expect the

M M potential function to be nearly pairwise additive. There is an upper bound on the ion concentration range within which it is sensible to compare the model with data for real systems if the pairwise addition approximation is made."

Table Il Mean-force potential W^{{ j}(r) of ion—ion interaction in dilute solutions of symmetrical electrolytes

The mean-force potentials of this table are special formulae of the general charge distribution.

Appendix B,for a single charge (ion).

, 1000 NAe I ^

x (Si-units) = — — x T ; T(ional concentration) = £ (a cj) A

EQEIC 1 j

e0: permittivity of vacuum; k: Boltzmann constant; e0: charge of proton; NA: Avogadro number;

WJ mean-force potential of the short-range forces.

Region Mean-force potential r ^ a oo

a < ^r< ^R * ^{+ W 1 S}

4n E0E r 4rc E0G t -f xR 1

r i R ^xlxe » p [ x ( R - r ) ]

4rc E⁰E r 1 + xR

A multitude of MM-Ievel HamiItonian models can be found for the same system.

The features of our chemical model are given in Refs. ³*⁴«^{7 2 )}.

i) The distance parameters a (minimum distance of two ions) and R (upper limit of the structured region around an ion) are fixed by chemical evidence.

ia) The lower distance is fixed as a hard-core radius by the center-to-center distance of the ions where these exist (e.g. alkali halides) or else is calculated from bond lengths or van der Waals volumes (e.g. tetraalkylammonium salts). For unsymmetrical ions like M e²B u²N⁺ or C²H⁵O " the shortest possible distance is taken to be the distance of closest approach. Some ions, e.g. L i⁺ in water or protic solvents, require the inclusion of a functional group of the solvent molecule (here: O H ) into the distance of closest approach. This leads to structures like L i⁺( R O H ) C l " with a = a^{c r y s t} + d^{O H}.

ip) The upper distance R is obtained by adding the length of one or more orientated solvent molecules to the distance of closest approach: R = a + ns, n = 1, 2, ...

Values a and s are quoted in Appendix A-2.

ii) The mean-force potentials O f W^{i j} (Eq. 14) are split into two parts representing Coulombic ion-ion interaction, Wf^j1, and short-range interactions, W J . A further subdivision of the WJ's which specifies contributions from induction, dispersion and chemical forces (e.g. H—bonding) is possible. A t the current stage of investigation, the contributions Wf^j1 for every region are obtained from the resolution of a set of Poisson-equations and appropriate boundary con- ditions and the W J are chosen as step potentials.

iii) Extensions

iiioc) A subdivision of the region a ^ r ^ R is useful when more than one shell of solvent molecules is orientated.

iiiP) The introduction of local permittivities is possible.

iiiy) The spherical charge symmetry can be replaced by an arbitrary charge distribution leading to angular-dependent potentials. Appendix B gives a summary of the appropriate potentials. Chemical kinetics makes use of this type of extension when kinetic salt, solvent, and substituent effects are treated for reactions between particles with complex charge distributions (see Sect. VIII).

4.3 The Ion-Pair Concept

Models of the electrolyte solution allow the introduction of the association concept if a critical distance around the central ion can be defined within which pair configurations of oppositely charged ions are considered as ion pairs. The link between the model and the experimentally determined thermodynamic property of the solution is an integral expression which can be subdivided in various ways

r g j j dr = r² exp WW"

kT dr -f- r² exp Wft''"

kT

d r . (17)

The choice of R is arbitrary within reasonable limits and then divides up the thermodynamic excess function, |Xy = v R T l n y^±, into contributions from the so- defined ion pair (degree of dissociation, a) and from the Tree' ions (activity coefficient of the free ions, y'^±), cf. E q . (7). Onsager characterized the situation as follows ^{7 3 >}: ⁴T h e distinction between free ions and associated pairs depends on an arbitrary convention. Bjerrum's choice is good, but we could vary it within reason.

In a complete theory this would not matter; what we remove from one page of the ledger would be entered elsewhere with the same effect."

Theory alone cannot provide a criterion for the best association constant. How- ever, the variety of solution models leads usually to more or less satisfactory association constants when all of the relevant experimental data including their dependence on temperature and pressure are considered. Once more, chemical evidence is a good criterion for the selection of the appropriate model. It will be shown in the following sections that the identification of the "critical distance of association"

with the cut-off distance of the short-range forces, R, in our chemical model yields association constants which are. almost independent of the experimental method of their determination.

The association concept is based on the equilibrium of Tree' ions and ion pairs in the solution

C^{z +} + A^z~ <± [C^{z +} A^z~ p . (18)

The concentration equilibrium constant of ion-pair formation, K^c — K^Ay '^{± 2} (see Eq. (8)), can be written ^{4 )} as shown in the first equation of Fig. 3 when using the reduced partition functions Q^p, Q ⁺, and Q_ of ion pairs, free cations and free anions.

As a first approximation the free cations and anions are considered as charged spheres of masses m⁺ and m _ . The ion pair is represented by an uncharged

12TrmjkT)312 Jnt

kC=1000NA ^U2- e* P l-ffi> OI = ' " " ^ ' / .z}'»-. AE0Ir) = Jr)

ZⁱS^t= ZⁱP^t=

1

.

a 4 r ^ R

-2qkT -2qkT

KA=WOOtvNA C²Bxpt^r^Jdr

R= q R=Aa

Prue Bjerrum contact pairs

Jnt _ ^rot ^vjb7

/ " / / i "

appropriate K^A

Q=R^Q^ R¹^ ^^Rf-1 P¹= R K- =^⁹P¹- • K"!^i}

VV*/'7 # O only if R^1 £ r £ RJ

^ l d r kT I

K^A = WOOtvNa I ^r2^exp 1²½ . ^ l d r kT Pi-1 ^J

KA= WOOtvNA

a.

Fig. 3. The family tree of association constants

particle of reduced mass, rh^p (Iii^p"¹ = m ^¹ + m l¹) , in a spherical box of radius R, meaning that an ion pair is formed if two oppositely-charged ions have approached to within a distance smaller than R. The difference in energies of the reacting species, A E0, can be identified with the mean-force potential W+ _ (a ^ r ^ R), Table II.

Finally, cf. Fig. 3, the equation

is obtained by this approximation. The activity coefficient of the dissociated part of the electrolyte, y '^±, is given by the relationship 3 , 6 5 )

y^± = exp

L

KRJ

(19c) Figure 3 shows the family tree of some association constants which can be found in the literature and indicates the presuppositions for deducing them from the initial equation. For example, Bjerrum's association constant^{5 )} and its appropriate activity coefficient are obtained from Eqs. (19) by setting R = q and W* _ = 0. As a further

example, the subdivision of the region a ^ r ^ R into a < R¹ < ... < R ⁿ < R is straightforward ^{4 )}. A current application is the case of stepwise ion-pair forma- tion ^{7 4}~^{7 6 )}, e.g.

H H H C d^{2 +} (aq) + S0²~(aq) «± C d^{2 +}( / O S O² ^ C d^{2 +} O ' S O i ^ C d^{2 +}+ S O² \

I I i ^ ^ I I N r I 1 I

region of free r = R = a + 2s ^r= R j = a + s r = R⁰ = a ions in

ions r ^ R contact (20)

The distance parameter a = a⁺ + a _ has its usual meaning, s is the length of an orientated solvent molecule.

Association constants, as determined by thermodynamic or transport process mesurements, are the basis for determining the short-range forces around the ions and mean-force potentials in dilute solutions. These experiments provide distance para- meters, R, as well as K^A-values and thus permit determination of short-range inter- action potentials, W^SJ _, via Eq. (19a), or an equivalent expression from Fig. 3 ³>⁴-^{7 2}-

7 4 - 8 1 ). On the other hand, molar quantities A G J = N^AW⁵J _ can be interpreted as the

non-Coulombic part of the Gibbs energy of the ion-pair formation reaction 3 4 , 7 4 •^{7 5 )}. In addition, calorimetric measurements and studies of the temperature-dependence of conductance provide enthalpies and entropies of ion-pair formation which yield valuable information. For example, alkali metal, alkaline earth and other divalent cation salts in protic solvents yield positive A H J and highly positive ASJ values in contrast to tetraalkylammonium salts which show A H * < O and small entropies, A S *^{4 , 7 4}~^{7 7 )}. Ion-pair formation within the former groups of salts involves the rearrangement of the solvation shells whereas that of the latter one scarcely docs at all. For more details on comprehensive series of measurements see Refs. ^{4 7 7 )}.

V Thermodynamic Properties of Electrolyte Solutions 5.1 Generalities

Properties E (c; p,T) of dilute and moderately concentrated electrolyte solutions (concentration range xq < 0.5; x (Table II), q (Eq. 19 b)), e.g. thermodynamic pro- perties Z , can be represented by a set of equations ^{7 2 )}

E(c; p, T) = E^{0 0}(p, T) + E'(ac, R ; p, T) (21 a)

l - o c 1

, ,>; y. - ^xp

ore y,"

where E^r (p, T) is the corresponding property of the infinitely dilute solution. The other symbols have the meaning given in the preceding sections. Transport properties are controlled by a similar set of equations, sec Section VI.

qx (21b, c)

The chemical model allows the determination of both values of E^{c c}(p, T) by a well- founded extrapolation method and values of R and Wf. independently of the special thermodynamic or transport property which is being investigated and thus provides data for other ones ³-^{7 2}-^{7 7}'^{8 1} "^{8 3}>. This fact can be used for computer methods which require the storing of a minimum of basic data to make these properties available.

The data bank E L D A R^{8 3 )} works on this principle. Spectroscopic or kinetic investi- gations do not necessarily furnish the complete set of parameters for establishing function E ' , Eq. (21a), see Sections VII and VIII.

5.2 Solution and Dilution Experiments

The basic equations of type Eq. (21a) needed for solution and dilution experiments are obtained from Eqs. (12 a, b). The apparent molar quantity O^z may be split into two parts, one for the Tree' ions of the chemical model, <J>^Z(FI) = O^z(r > R), the other one for the ion pairs, O^z(IP) = O^z( a ^ r g R ) . This assumption leads to the relationship

O^z - O? = cD^z<¹ = IacD^z(FI) + (1 - a) CD^z(IP)] - O? (22) which is introduced into Eq. (12a) yielding the equation

^ = A^{S O L}Z ? + <xO^ze,(FI) + (1 - a) A Z^a . (23) n^Y

cD^zcl is the corresponding relative apparent molar quantity. For symmetrical electro- lytes at moderate and low concentrations the quantity O^{z e l}(IP) equals A Z^A =

cDz(IP) — Og, i-^e- ^{t ne} niolar quantity for the formation of an ion pair from its cation and anion which are initially infinitely separated.

As an example, heat of dilution experiments and the information they provide about solution properties will be discussed. A comparison of Eqs. (12a, 22) and Fig. 1 shows that (DH¹ (in the literature^sometimes called cD^L) and the measured negative heat of dilution are identical, i.e.

Off¹ = - A^{d n}H^{0 0} . (24)

Hence, when diluting an electrolyte solution at molality m to a molality m' by adding an appropriate amount of solvent, the accompanying heat of process is

AcD^rH^el = O f f V O - Off¹(Hi) = Ot^rOff¹(FI) - OtOff¹(FI) + (a - a') AH° . (25 a) Using the well-known thermodynamic relationship

m

Off¹(Fl) = - v R T^{2 ~}

J (^

-)

<

>

Unlv.-Bibliofhek Regensburg

and the activity coefficient of the free ions required by the chemical model, y '^±, after conversion of y'^± (Eq. 7) into the molal scale, the following theoretical expression ^{7 7 )} 'is obtained

cDffi(FI) = - v R T² 9 In 6 6T ⁺ T

xq

1 + xR 3 j (25c)

where the function er(xR) is given by the relationship

G(yR) = - 4^ 3 [1 + x R - — ^ — - 2 In (1 + x R ) ] . (25 d) (xR)"* 1 + x R

a is the cubic expansion coefficient of the solvent; the other symbols where already defined in the preceding text.

The data analysis of dilution measurements with the set of Eqs. (25) yields the basic quantities K ^a and R of the chemical model and A H^a (heat of ion-pair formation).

The entropy of ion-pair formation, A S^a , can be calculated from K ^a and A H^a in the usual way. Table III shows examples of a simultaneous determination of K^A,

R^{E X P} and A H^{0 A}.

The association constants of table III can be compared with those from conductance measurements, K j ^ , and are found to be in perfect agreement, e.g. Kj^(MgSO⁴ZH²O)

= 160 d m³ mol ~¹. The agreement of the R^{e x p}-values of Table III for aqueous solutions with those of the ion-pair model, E q . (20), should be stressed as an important result.

The calculated values, R^{c a u}. , correspond to R = a + 2s (here s = d⁰^ dimension of O H ) according to this model. The agreement of R^{e x p} with BjerrunVs distance para- meter q, which is often used as the upper limit of association and which depends only on the permittivity of the solvent [cf. E q . (19 b)], is less satisfactory. For aqueous solutions of 2,2-electrolytes at 25 ⁰C q equals 1.43 nm, independent of the ionic radii.

The same situation is given for non-aqueous solutions, e.g. propanol solutions in Table III. The association constants from calorimetric and conductance measurements

Table III Thermodynamic quantities of ion-pair formation in water and propanol (25 ¹C ) from measurements of heats of dilution ^{7 7 )}

Solvent Electrolyte A H0 A D

calc/nm dm³ mol ¹ J m o l "¹ nm •• - "

a + 2 d^{o l l} a + s q

H²O M g S O⁴ 16-1 5780 0.93 0.88 1.43

C a S O⁴ 192 6670 0.95 0.91

C d S O⁴ 239 8390 0.96 0.91

N i S O⁴ 210 5440 0.91 0.88

C³H⁷O H Nal 206, 18930 0.97 0.88 1.01 1.37

KI 374 19060 0.99 0.91 1.04

RbI 527 17550 0.95 0.93 1.06

agree satisfactorily ^{7 7 )}. The experimentally determined distances, R^{e x p}, differ distinctly from q, which equals 1.37 nm for all propanol solutions of 1,1-electrolytes at 25 °C.

Two values of R^{c a l c} are quoted in Table III, one for the configuration C ⁺ ( O H ) ( O H ) A^-, i.e. R^{c a l c} = a + 2 d^{O H}, the other for C ⁺ (propanol)A~, i.e. R^{c a i c}

= a + s (cf. Appendix A-2). Both values R^{c a l c} are compatible with the experimental value, R^{e x p}.

Comparing Table III with the results of conductance data at various temperatures shows a further important feature. The temperature-dependence of conductance data yields AHjJ-values via the relationship

which agree with the AH^A-values from calorimetric measurements, i.e. from conduct- ance measurements on solutions of NaI in propanol a value of A H^A = 18 8 0 0 J mol ~¹ is determined.

The second example concerning heat of solution measurements was chosen to stress a crucial problem in non-aqueous electrochemistry. This is the proper extrapolation to infinite dilution when association of the electrolyte occurs ^{8 4}~^{8 7 )}.

Figure 4 shows that the validity range of the limiting law is attained only at very low concentrations (here < 1 0 ~⁵M ) , generally inaccessible to measurements. Hence, extrapolation from measured values (>5 • I O "³ M ) yields erroneous data. Reliable

(26)

0.06

C "2

Fig. 4. Relative apparent molar enthalpy of KI in propanol (25 °C) from heat of dilution measurements, (a) measured curve;

(b) limiting law; for explanation see text

Table IV Heat of solution data of NaI in 1-propanol at 25 ⁰C ^{8 4 )}

m • IO³ A^{S O 1}H^Y W^{1 AS O 1HY}

J mol"¹ mol k g^{- 1} J m o l "¹ J m o l^{- 1}

AS O 1HY

J mol"¹

20.56 -19090 8898 -27988

21.70 -18850 9024 -27874

22.44 -18840 9103 -27943

22.64 -18780 9123 -27903

23.70 -18700 9250 -27950

Mean: -27932

extrapolations require the use of Eq. (23) and an appropriate model. The Debye- Huckel limiting law and its empirical extensions are generally insufficient-

Table IV shows A^{S O 1}H^Y, ^ H^{1 an (}i extrapolated A^{S O L}HY-values as a function of molality for solutions of NaI in n-propanol ^{8 4 )}. The <X>H'-values^were determined from the dilution measurements of Table III. Taking into account that <&[f contributes about 50 % of A^{S O 1}H^y, it is obvious that reliable A^{s o l}Hy-Values can only be obtained from theoretically sound extrapolation methods. For comparison, Abraham et al. ^{8 5 )} estimated a value of A^{s o l}H y = —23510 J m o l^{- 1} from their measurements on solutions of NaI in propanol. A re-evaluation of these measurements based on the chemical model and Eq. (23) yields A^{S O 1}H^{Y 1} = —27200 ± I s o j m o r^lJ n s a t i s f a c t o r y agreement with our values in Table I V .

A n example of Eq. (23) for determining molar volumes is given in Ref. ^{8 1} \ Compar- ing the results of molar volume measurements with the pressure dependence of the association constant from conductance experiments shows satisfactory agreement, i.e. the equation

d In K^A\ A V^A

(27)

dp Jj R T

is fulfilled when K^A-values from pressure-dependent conductance and AV^A-values from density measurements are combined.

5.3 EMF-Measurements

For AG-measurements, emf or solubility products, the chemical potential, Eqs. (6, 7), is an appropriate form of the basic Eqs. (21). For example, the emf of the galvanic cell without liquid junction

A g / A g C l ( s ) / K⁺C l ~ (sol; concentration c mol d m "³) / ( K , Hg) is given by the set of equations

2 RT

E = E^{0 ;}- In(Otylc) (28a)

F 1 - Of 1

KA ^ — - n : y± = exp

Ot²C y -

xq U x R

(28 b, c)

Emf measurements yield reliable standard potentials E⁰ only when data analysis uses well-founded extrapolation methods which take into account association of the electrolyte compounds ^{8 8 )}. A knowledge of reliable standard potentials is important for electrochemistry in non-aqueous solutions, especially for solvation studies and technological investigations. A comprehensive survey of these questions is in preparation. Data analysis with the help of Eqs. (28) gives K ^a and R values which are compatible with those from other methods. Table V illustrates the satisfactory agreement of activity coefficients from emf measurements, y^±(emf), and heats of dilution, y±( O HC L) , both evaluated by appropriate methods.

Table V Activity coefficients of aqueous C d S O⁴ solutions at 25 ⁰C

IO⁴C 10.0 30.0 50.0 100.0

mol d m^{- 3}

y* (0),71) 0.699 0.551 0.476 0.383

y^±i c m 0 0.698 0.552 0.481 0.388

5.4 Some Remarks on Thermodynamic Investigations

The possibility of determining the quantities K^A, A Z^a and R controlling association as well as the values of u^ and A^{s o l}Z y relevant for solvation gives a convincing reason for making comprehensive measurements on the properties of dilute solutions. Temperature- and pressure-dependent data in particular are needed at the present time. The determination of thermodynamic quantities has been the object of numerous investigations. However, the examples given illustrate the difficulties in getting standard values for the thermodynamic properties of partially associated electrolytes. This may be the reason why most of the work on the thermodynamic properties of electrolyte solutions, such as measurement of the apparent molal volumes ^{8 9} "^{9 6 )}, apparent molal heat capacities ^{8 9}-^{9 0}'^{9 7}) and heats of solution ^{9 8}-^{1 0 0}> , has been concerned with solutions of non-associated electrolytes. In this case the extrapolation to infinite dilution is carried out with the help of E q . (12a) in combination with the Debye-Hiickel activity coefficient or its extended forms ^{1 0 1}"^{1 0 4 )}.

The features of the chemical model are well-founded extrapolations towards Zy and related quantities on the one hand and the generation of a basic set of model parameters R and W^{i j} (via K^A) independent of the special experimental method on the other hand. Moreover, the distance parameter R is always found to be in accordance with the dimensions of a configuration of ions and orientated adjacent solvent molecules which is compatible with general chemical evidence. A s far as chemical models are McMillan-Mayer-level Hamiltonian models, they permit the use of statistical thermodynamic relationships for calculating the solvent properties in a well-founded manner.

It should be mentioned in this context that investigations on dilute solutions, from which reliable information is expected, require precise measurements down to very

low concentrations (c ~ I O^{- 4} mol d m^{- 3}) . Very sensitive and precise apparatus is need- ed, see Refs.⁴-⁷⁷'⁷⁹-⁸¹-1 0 5 - 1 0 7 ^ a n (j p^{U r}i f^{l c a}t i o n of solvents and solutes together with the purity control of solvents, solutes and solutions are often the major part of these investigations.

5.5 Ion Solvation

Ion solvation is the transfer process of the separated ions of a pure electrolyte compound Y from the vacuum to the infinitely dilute solution in a solvent S.

In the case of ionophoric electrolytes the solvation quantities A^{s o l v}Z y are related to the corresponding solution quantities, A^{s o l}Z y (Eq. (12 b)), via lattice quantities,

A^{l a t}Z y according to Fig. 1.

Lattice energies of many electrolytes are known ^{1 0 8} "^nl) and in combination with the experimentally determined solution energies yield the solvation energies and related quantities A^{s o l v}Z y . For tables of solution data see Refs. ^{3 6}-^{n 2}- ^{U 3 )}.

Besides the solvation quantities, transfer quantities, A^{t r}Z y , can be advantageously used, cf. Fig. 1. They give an account of the change in Z when the electrolyte Y is transferred from solvent S' to solvent S. With water as the reference solvent, S', the transfer activity coefficients, ^myY 1 1 4 _ 1 1 6^ are obtained from the Gibbs transfer energy, A^{t r}G y , by the relationship

^^{t r}G ? = A^{s o l v}G » ( S ) - A^{s o l v}G - ( W ) = R T In ^my^Y . (29) The choice of the appropriate concentration scale for standard thermodynamic

functions of transfer was extensively discussed by Ben-Naim ^{1 1 7 )} who showed that the molarity scale has a number of advantages over the others.

Separation into ionic transfer activity coefficients for the electrolyte C^{7 +} A⁷" is executed with the help of the equation

my²Y = (^mY⁺) (^mr - ) - (30)

The interest of theory and technology in single ion solvation and transfer quantities originates in their importance for solution structure, kinetic, analytical or surface problems, i.e. for all problems involving the solvation shell or its rearrangement. For example, transfer proton activity coefficients ^my^H+ are used for transferring the pH-scale from water to other solvents:

pH(S) = p H ( W ) + l o g^mY^{H +}. (31)

Further examples of technical importance are found in the field of extraction processes ^{1 1 8}\ ionic equilibria and emf measurements ^{1 1 9}\ and analytical applications

1 2 0 . 1 2 1 )

The requirements of theory both for solvation and transfer data of single ions are similar. A complete theory would require the knowledge of all molecular distribution functions and mean-force potentials between the ions and the solvent molecules. As already stressed in Section II such a theory is unavailable with the present state of knowledge. In the endeavour to represent solvation by models, the